To simplify notation we set \(y (y_1,\ldots ,y_k)^{\prime }\)  for the observed effect sizes and \(\sigma ^2 (\sigma _1^2,\ldots ,\sigma _k^2)^{\prime }\)  for the respective sampling variances, pooled from our \(k\)  studies. Given an improper uniform or normal prior for the effect \(\mu \) , the effect's conditional posterior distribution for a given heterogeneity \(\tau ^2\) , \(p(\mu \mid \tau ^2,\sigma ^2, y)\) , is again normal [1]}. In case of an improper uniform prior we obtain \(\hat{\mu } = \frac{\sum _{i=1}^k w_i y_i}{\sum _{i=1}^k w_i}\)  and \(\hat{\sigma } = 1/\sqrt{\sum _{i=1}^k w_i}\)  as posterior estimators for the mean and the standard deviation, respectively, where \(w_i = (\sigma _i^2 + \tau ^2)^{-1}\) . In case of a normal prior for \(\mu \)  with mean \(\mu _p\)  and variance \(\sigma _p^2\)  we obtain posterior mean \(\hat{\mu } = \frac{\mu _p/\sigma _p^2 + \sum _{i=1}^k w_i y_i}{\sigma _p^{-2} + \sum _{i=1}^k w_i}\)  and standard deviation \(\hat{\sigma } = 1/\sqrt{\sigma _p^{-2} + \sum _{i=1}^k w_i}\)  [2]}, [1]}.
